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Interval exchange transformations and low-discrepancy

  • Christian Weiß
Article
  • 12 Downloads

Abstract

In Masur (Ann Math 115(1):169–200, 1982) and Veech (J Anal Math 33:222–272, 1978), it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the question under which conditions the orbits yield low-discrepancy sequences. The case of \(n=2\) intervals corresponds to circle rotation, where conditions for low-discrepancy are well-known. In this paper, we give corresponding conditions in the case \(n=3\). Furthermore, we construct infinitely many interval exchange transformations with low-discrepancy orbits for \(n \ge 4\). We also show that these examples do not coincide with LS-sequences if \(S \ge 2\).

Keywords

Low-discrepancy Interval exchange transformation Uniform distribution Kronecker sequences LS-sequences 

Mathematics Subject Classification

11B50 11B83 11K38 37E10 

Notes

Acknowledgements

The author thanks the anonymous referees for their useful comments.

References

  1. 1.
    Carbone, I.: Discrepancy of \(LS\)-sequences of partitions and points. Ann. Mat. Pura Appl. 191, 819–844 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carbone, I., Iacò, M., Volčič, A.: A dynamical systems approach to the Kakutani–Fibonacci sequence. Ergod. Theory Dyn. Syst. 34, 1794–1806 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Drmota, M., Tichy, R.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer, Berlin (1997)Google Scholar
  4. 4.
    Grabner, P., Hellekalek, P., Liardet, P.: The dynamical point of view of low-discrepancy sequences. Unif. Distrib. Theory 7(1), 11–70 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kakutani, S.: A problem on equidistribution on the unit interval \([0,1[\). In: Measure Theory (Proceedings of the Conference Oberwolfach, 1975), Lecture Notes in Mathematics, vol. 541 (pp. 369–375). Springer, Berlin (1975)Google Scholar
  6. 6.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)zbMATHGoogle Scholar
  7. 7.
    Larcher, G.: Discrepancy estimates for sequences: new results and open problems. In: Kritzer, P., Niederreiter, H., Pillichshammer, F., Winterhof, A. (eds.) Uniform Distribution and Quasi-Monte Carlo Methods, Radon Series in Computational and Applied Mathematics, pp. 171–189. DeGruyter, Berlin (2014)Google Scholar
  8. 8.
    Masur, H.: Interval exchange transformations and measured foliations. Ann. Math. (2) 115(1), 169–200 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods, Number 63 in CBMS-NSF Series in Applied Mathematics. SIAM, Philadelphia (1992)CrossRefGoogle Scholar
  10. 10.
    Schmidt, W.M.: Irregularities of distribution VII. Acta Arith. 21, 45–50 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Steinig, J.: A proof of Lagrange’s theorem on periodic continued fractions. Arch. Math. 59, 21–23 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Veech, W.A.: Interval exchange transformations. J. Anal. Math. 33, 222–272 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Viana, M.: Ergodic theory of interval exchange maps. Rev. Mat. Complut. 19(1), 7–100 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Volčič, A.: A generalization of Kakutani’s splitting procedure. Ann. Mat. Pura Appl. (4) 190(1), 45–54 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Weiß, C.: On the Classification of LS-sequences, arXiv:1706.08949 (2017)
  16. 16.
    Yoccoz, J.-C.: Continued fraction algorithms for interval exchange maps: an introduction. In: Frontiers in Number Theory, Physics, and Geometry I, pp. 401–435. Springer (2006)Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hochschule Ruhr WestMülheim an der RuhrGermany

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